Gas gauge reads full all the time

it will help if you tell us what kind of car it is and dose this problem happen after you fill it while the car is running or is it off ?

when the tank is full the resistance is low so the power through the gage make is read full
as the tank empties , the resistance increases so the less current means the gauge reads less until empty there is little or no power , the gauge reads empty
simple check , turn the ignition off, no power , no gauge reading

Bad fuel sending unit.

SOUND LIKE GAUGES IS FAULTY.

your gauge is part of the integrated unit in the fuel pump module

sending unit on fuel pump is bad, the sending unit shorts out gauge will go to empty or full, never read right, turn on switch ,it goes to full or empty, ,check the pump,hope this helps,thanks robert b.

I am very familiar with this problem, u will need to replace the gas gauge sending unit in the fuel tank.

Do the doors unlock when you shut the key off? Also unhook the purple wire to the sending unit in the tank, Gauge should read empty. Ground the purple wire, it should read past full. If it works this way, replace the sending unit in the tank.

This is a normal concern.
The instrument cluster uses 4 different operating modes to calculate the fuel level:
Anti-slosh (default mode)
Key OFF fueling
Key ON fueling
Recovery
After a fuel fill up, the time for the fuel gauge to move from empty (E) to full (F) ranges from 2 seconds to 55 minutes depending on which operating mode the fuel gauge is in.
The default fuel gauge mode is called the anti-slosh mode. To prevent fuel gauge changes from fuel slosh (gauge instability due to changes in fuel sender readings caused by fuel moving around in the tank), the fuel gauge takes approximately 55 minutes to go from empty (E) to full (F).
The key OFF fueling mode (2 seconds to read empty [E] to full [F]) requires 3 conditions be met:
The key must be in the OFF position throughout the entire refueling of the vehicle.
At least 15% of the vehicle's fuel capacity must be added to the fuel tank.
The instrument cluster must receive a valid key ON fuel sender reading within 1 second of the key being put into the RUN position. The key ON sample readings are considered valid if the fuel sender reading is between 15 ohms ± 2 ohms and 160 ohms ± 4 ohms.
If these conditions are not met, the fuel gauge stays in the anti-slosh mode, which results in a slow to read full (F) event.
The key ON fueling mode (approximately 90 seconds to read empty [E] to full [F]) requires 3 conditions be met:
The transmission is in PARK (P).
The key is in the RUN position.
At least 15% of the vehicle's fuel capacity must be added to the fuel tank.
In key ON fueling mode, a 30-second timer activates after the transmission is put into the PARK (P) position (automatic transmissions). When the 30-second time has elapsed and at least 15% of the vehicle's fuel capacity has been added, the fuel gauge response time is 90 seconds to read from empty (E) to full (F). When the transmission is shifted out of PARK (P), the fuel gauge strategy reverts to the anti-slosh mode. The key on refueling mode prevents slow to read full events from happening if the customer refuels the vehicle with the key in the RUN position.
Recovery mode is incorporated into the instrument cluster strategy to recover from a missing fuel level input during a refueling event. Missing fuel level inputs result from intermittent opens in the fuel sender or its circuits. Recovery mode (empty [E] to full [F] approximately 20 minutes) is initiated when the following 2 conditions are met:
The instrument cluster is in the anti-slosh (default) mode.
The actual fuel level in the tank is greater than what is being displayed by the fuel gauge

Sounds to be a faulty tank sender unit, these senders use a fine coiled wire on which a brush slides along to vary its resistance.
If the coiled wire breaks midway, the only readings will be full or empty.
I think sender unit is mounted to internal fuel pump on this model (top of fuel tank)
Hope this helps, regards